# Non-Forced-Choice Models (feasible outside option)¶

## Utility Maximization with an Outside Option¶

### Strict¶

A general choice dataset $$\mathcal{D}$$ on a set of alternatives $$X$$ is explained by (strict) utility maximization with an outside option if there is a strict linear order $$\succ$$ on $$X$$ and an alternative $$x^*\in X$$ such that for every menu $$A$$ in $$\mathcal{D}$$

$\begin{split}C(A) = \left\{ \begin{array}{ll} \mathcal{B}_{\succ}(A), & \text{if x\succ x^* for \{x\}= \mathcal{B}_\succ(A)}\\ &\\ \emptyset, & \text{otherwise}\\ \end{array} \right.\end{split}$

where

$\mathcal{B}_{\succ}(A):=\Big\{x\in A: x\succ y\; \text{for all y\in A\setminus\{x\}}\Bigr\}$

is the strictly most preferred alternative in $$A$$ according to $$\succ$$.

### Non-strict¶

A general choice dataset $$\mathcal{D}$$ on a set of alternatives $$X$$ is explained by (non-strict) utility maximization with an outside option if there is a weak order $$\succsim$$ on $$X$$ and an alternative $$x^*\in X$$ such that for every menu $$A$$ in $$\mathcal{D}$$

$\begin{split}C(A) = \left\{ \begin{array}{ll} \mathcal{B}_{\succsim}(A), & \text{if x\succ x^* for all x\in \mathcal{B}_\succsim(A)}\\ &\\ \emptyset, & \text{otherwise}\\ \end{array} \right.\end{split}$

and

$x\sim y\;\; \text{for distinct}\; x,y\; \text{in}\; X$

where

$\mathcal{B}_{\succsim}(A):=\{x\in A: x\succsim y\; \text{for all y\in A}\}$

is the set of weakly most preferred alternatives in $$A$$ according to $$\succsim$$.

### Strict¶

A general choice dataset $$\mathcal{D}$$ on a set of alternatives $$X$$ is explained by (strict) overload-constrained utility maximization if there is a strict linear order $$\succ$$ on $$X$$ and an integer $$n$$ such that for every menu $$A$$ in $$\mathcal{D}$$

$\begin{split}C(A) = & \left\{ \begin{array}{ll} \mathcal{B}_{\succ}(A), & \text{if |A|\leq n}\\ &\\ \emptyset, & \text{otherwise} \end{array} \right.\end{split}$

where

$\mathcal{B}_{\succ}(A):=\Big\{x\in A: x\succ y\; \text{for all y\in A\setminus\{x\}}\Bigr\}$

is the strictly most preferred alternative in $$A$$ according to $$\succ$$.

### Non-strict¶

A general choice dataset $$\mathcal{D}$$ on a set of alternatives $$X$$ is explained by (non-strict) overload-constrained utility maximization if there is a weak order $$\succsim$$ on $$X$$ and an integer $$n$$ such that for every menu $$A$$ in $$\mathcal{D}$$

$\begin{split}C(A) = & \left\{ \begin{array}{ll} \mathcal{B}_{\succsim}(A), & \text{if |A|\leq n}\\ &\\ \emptyset, & \text{otherwise} \end{array} \right.\end{split}$

and

$x\sim y\;\; \text{for distinct}\; x,y\; \text{in}\; X$

where

$\mathcal{B}_{\succsim}(A):=\{x\in A: x\succsim y\; \text{for all y\in A}\}$

is the set of weakly most preferred alternatives in $$A$$ according to $$\succsim$$.

## Incomplete-Preference Maximization: Maximally Dominant Choice¶

### Strict¶

A general choice dataset on a set of alternatives $$X$$ is explained by (strict) maximally dominant choice if there is a strict partial order $$\succ$$ on $$X$$ such that for every menu $$A$$ in $$\mathcal{D}$$

$\begin{split}C(A) = \left\{ \begin{array}{ll} \mathcal{B}_{\succ}(A), & \text{if \mathcal{B}_\succ(A)\neq\emptyset}\\ &\\ \emptyset, & \text{otherwise}\\ \end{array} \right.\end{split}$

where

$\mathcal{B}_{\succ}(A):=\Big\{x\in A: x\succ y\; \text{for all y\in A\setminus\{x\}}\Bigr\}$

is the (possibly non-existing) strictly most preferred alternative in $$A$$ according to $$\succ$$.

### Non-strict¶

A general choice dataset $$\mathcal{D}$$ on a set of alternatives $$X$$ is explained by (non-strict) maximally dominant choice if there is an incomplete preorder $$\succsim$$ on $$X$$ such that for every menu $$A$$ in $$\mathcal{D}$$

$\begin{split}C(A) = \left\{ \begin{array}{ll} \mathcal{B}_{\succsim}(A), & \text{if \mathcal{B}_{\succsim}(A)\neq\emptyset}\\ &\\ \emptyset, & \text{otherwise}\\ \end{array} \right.\end{split}$

and

$x\sim y\;\; \text{for distinct}\; x,y\; \text{in}\; X$

where

$\mathcal{B}_{\succsim}(A):=\{x\in A: x\succsim y\; \text{for all y\in A}\}$

is the (possibly empty) set of the weakly most preferred alternatives in $$A$$ according to $$\succsim$$.

## Incomplete-Preference Maximization: Partially Dominant Choice (non-forced)¶

A general choice dataset $$\mathcal{D}$$ on a set of alternatives $$X$$ is explained by partially dominant choice (non-forced) if there exists a strict partial order $$\succ$$ on $$X$$ such that for every menu $$A$$ in $$\mathcal{D}$$ with at least two alternatives

$\begin{split}\begin{array}{llc} C(A)=\emptyset & \Longleftrightarrow & x\nsucc y\;\; \text{and}\;\; y\nsucc x\;\; \text{for all}\;\; x,y\in A\\ & &\\ C(A)\neq\emptyset & \Longleftrightarrow & C(A)= \left\{ \begin{array}{lll} & & \hspace{-12pt} z\nsucc x\qquad \text{for all}\;\; z\in A\\ x\in A: & & \;\;\;\;\;\;\text{and}\\ & & \hspace{-12pt} x\succ y\qquad \text{for some}\;\; y\in A \end{array} \right\} \end{array}\end{split}$

Note

In its distance-score computation of this model, Prest penalizes deferral/choice of the outside option at singleton menus. Although this is not a formal requirement of the model, its predictions at non-singleton menus are compatible with the assumption that all alternatives are desirable, and hence that active choices be made at all singletons.